# A Tree of Numbers

**Number Theory**Level 4

Bob has the number 2014 written on a chalkboard. He splits the number into a sum of two integers, and writes out the product of the two aside. Then he is left with a sum of two numbers. He then splits both of these two and lists out the product. When he can no longer split a number into a sum of two positive integers, he adds the products he had listed out. As an example, if Bob started with 10, he could split it as \[\begin{equation*}\begin{split} 10 & = 6 + 4 \\ & = (3 + 3) + 4\\ & = ((1+2) + (1+2)) + (2+2)\\ & = ((1 + (1+1)) + (1+(1+1))) + ((1+1) + (1+1)) \end{split} \quad \begin{split} 24\\ 9\\ 2, 2,4\\ 1,1,1,1\\ \end{split}\\ \implies 45 \end{equation*}\] Find the maximum possible value that Bob can get.

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